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Shakhmatov Dmitri - One of the best experts on this subject based on the ideXlab platform.
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Compactness properties defined by open-point games
2019Co-Authors: Dorantes-aldama Alejandro, Shakhmatov DmitriAbstract:Let S be a topological property of sequences (such as, for example, "to contain a Convergent Subsequence" or "to have an accumulation point"). We introduce the following open-point game OP(X,S) on a topological space X. In the n'th move, Player A chooses a non-empty open subet U_n of X, and Player B responds by selecting a point x_n in U_n. Player B wins the game if the sequence (x_n) satisfies property S in X; otherwise, Player A wins. The (non-)existence of regular or stationary winning strategies in OP(X,S) for both players defines new compactness properties of the underlying space X. We thoroughly investigate these properties and construct examples distinguishing half of them, for an arbitrary property S sandwiched between sequential compactness and countable compactness
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TWO OPEN-POINT GAMES RELATED TO SELECTIVE (SEQUENTIAL) PSEUDOCOMPACTNESS, WITH APPLICATION TO 1-CL-STARCOMPACTNESS PROPERTY OF MATVEEV (Research Trends on Set-theoretic and Geometric Topology and their cooperation with various branches)
'Research Institute for Mathematical Sciences Kyoto University', 2018Co-Authors: Dorantes-aldama Alejandro, Shakhmatov DmitriAbstract:A topological space X is selectively sequentially pseudocompact (selectively pseudocompact) if for every sequence {U_{n} : n in mathrm{N}} of non-empty open subsets of X, one can choose a point x_{n} in U_{n} for every n in mathrm{N} in such a way that the sequence {x_{n} : n in mathrm{N}} has a Convergent Subsequence (respectively, has an accumulation point in X). It was shown by the authors in [3] that the class of selectively sequentially pseudocompact spaces is closed under taking arbitrary products and continuous images, contains the class of dyadic spaces and forms a proper subclass of the class of selectively pseudocompact spaces. Moreover, the latter class coincides with the class of strongly pseudocompact spaces of García-Ferreira and Ortiz-Castillo [7]. In this paper, we define two topological games closely related to the class of selectively (sequentially) pseudocompact spaces. Let X be a topological space. At round n, Player A chooses a non-empty open subset U_{n} of X, and Player B responds by selecting a point x_{n} in U_{n}. In the selectively sequentially pseudocompact game Ssp(X), Player B wins if the sequence {x_{n} : nin mathrm{N}} has a Convergent Subsequence; otherwise Player A wins. In the selectively pseudocompact game Sp(X), Player B wins if the sequence {x_{n} : nin mathrm{N}} has an accumulation point in X; otherwise Player A wins. The (non-)existence of winning strategies for each player in the game Ssp(X) (in the game Sp(X)) defines a compactnesslike property of X sandwiched between sequential compactness (countable compactness) and selective sequential pseudocompactness (selective pseudocompactness) of X. We prove that a topological space X such that Player A does not have a winning strategy in Sp(X), is 1-cl-starcompact in the sense of Matveev. As an application of this result, we give an example of a locally compact, first-countable, zero-dimensional, 1-cl-starcompact space without a dense relatively countably compact subspace. This shows that Theorem 15 in Matveev's survey [10] is not reversible
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Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups
'Elsevier BV', 2017Co-Authors: Dorantes-aldama Alejandro, Shakhmatov DmitriAbstract:A space X is selectively sequentially pseudocompact if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in each U_n in such a way that the sequence (x_n) has a Convergent Subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion-free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topology. Since selectively sequentially pseudocompact spaces are strongly pseudocompact in the sense of Garc\'ia-Ferreira and Ortiz-Castillo, this provides a strong positive (albeit partial) answer to a question of Garc\'ia-Ferreira and Tomita
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Selective sequential pseudocompactness
'Elsevier BV', 2017Co-Authors: Dorantes-aldama Alejandro, Shakhmatov DmitriAbstract:We say that a topological space X is selectively sequentially pseudocompact (SSP for short) if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in U_n for every n in such a way that the sequence (x_n) has a Convergent Subsequence. We show that the class of SSP spaces is closed under taking arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by Garc\'ia-Ferreira and Ortiz-Castillo. We investigate basic properties of this new class and its relations with known compactness properties. We prove that every omega-bounded (=the closure of which countable set is compact) group is SSP, while compact spaces need not be SSP. Finally, we construct SSP group topologies on both the free group and the free Abelian group with continuum-many generators
Dorantes-aldama Alejandro - One of the best experts on this subject based on the ideXlab platform.
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Compactness properties defined by open-point games
2019Co-Authors: Dorantes-aldama Alejandro, Shakhmatov DmitriAbstract:Let S be a topological property of sequences (such as, for example, "to contain a Convergent Subsequence" or "to have an accumulation point"). We introduce the following open-point game OP(X,S) on a topological space X. In the n'th move, Player A chooses a non-empty open subet U_n of X, and Player B responds by selecting a point x_n in U_n. Player B wins the game if the sequence (x_n) satisfies property S in X; otherwise, Player A wins. The (non-)existence of regular or stationary winning strategies in OP(X,S) for both players defines new compactness properties of the underlying space X. We thoroughly investigate these properties and construct examples distinguishing half of them, for an arbitrary property S sandwiched between sequential compactness and countable compactness
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TWO OPEN-POINT GAMES RELATED TO SELECTIVE (SEQUENTIAL) PSEUDOCOMPACTNESS, WITH APPLICATION TO 1-CL-STARCOMPACTNESS PROPERTY OF MATVEEV (Research Trends on Set-theoretic and Geometric Topology and their cooperation with various branches)
'Research Institute for Mathematical Sciences Kyoto University', 2018Co-Authors: Dorantes-aldama Alejandro, Shakhmatov DmitriAbstract:A topological space X is selectively sequentially pseudocompact (selectively pseudocompact) if for every sequence {U_{n} : n in mathrm{N}} of non-empty open subsets of X, one can choose a point x_{n} in U_{n} for every n in mathrm{N} in such a way that the sequence {x_{n} : n in mathrm{N}} has a Convergent Subsequence (respectively, has an accumulation point in X). It was shown by the authors in [3] that the class of selectively sequentially pseudocompact spaces is closed under taking arbitrary products and continuous images, contains the class of dyadic spaces and forms a proper subclass of the class of selectively pseudocompact spaces. Moreover, the latter class coincides with the class of strongly pseudocompact spaces of García-Ferreira and Ortiz-Castillo [7]. In this paper, we define two topological games closely related to the class of selectively (sequentially) pseudocompact spaces. Let X be a topological space. At round n, Player A chooses a non-empty open subset U_{n} of X, and Player B responds by selecting a point x_{n} in U_{n}. In the selectively sequentially pseudocompact game Ssp(X), Player B wins if the sequence {x_{n} : nin mathrm{N}} has a Convergent Subsequence; otherwise Player A wins. In the selectively pseudocompact game Sp(X), Player B wins if the sequence {x_{n} : nin mathrm{N}} has an accumulation point in X; otherwise Player A wins. The (non-)existence of winning strategies for each player in the game Ssp(X) (in the game Sp(X)) defines a compactnesslike property of X sandwiched between sequential compactness (countable compactness) and selective sequential pseudocompactness (selective pseudocompactness) of X. We prove that a topological space X such that Player A does not have a winning strategy in Sp(X), is 1-cl-starcompact in the sense of Matveev. As an application of this result, we give an example of a locally compact, first-countable, zero-dimensional, 1-cl-starcompact space without a dense relatively countably compact subspace. This shows that Theorem 15 in Matveev's survey [10] is not reversible
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Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups
'Elsevier BV', 2017Co-Authors: Dorantes-aldama Alejandro, Shakhmatov DmitriAbstract:A space X is selectively sequentially pseudocompact if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in each U_n in such a way that the sequence (x_n) has a Convergent Subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion-free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topology. Since selectively sequentially pseudocompact spaces are strongly pseudocompact in the sense of Garc\'ia-Ferreira and Ortiz-Castillo, this provides a strong positive (albeit partial) answer to a question of Garc\'ia-Ferreira and Tomita
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Selective sequential pseudocompactness
'Elsevier BV', 2017Co-Authors: Dorantes-aldama Alejandro, Shakhmatov DmitriAbstract:We say that a topological space X is selectively sequentially pseudocompact (SSP for short) if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in U_n for every n in such a way that the sequence (x_n) has a Convergent Subsequence. We show that the class of SSP spaces is closed under taking arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by Garc\'ia-Ferreira and Ortiz-Castillo. We investigate basic properties of this new class and its relations with known compactness properties. We prove that every omega-bounded (=the closure of which countable set is compact) group is SSP, while compact spaces need not be SSP. Finally, we construct SSP group topologies on both the free group and the free Abelian group with continuum-many generators
Taubes, Clifford Henry - One of the best experts on this subject based on the ideXlab platform.
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The behavior of sequences of solutions to the Vafa-Witten equations
2017Co-Authors: Taubes, Clifford HenryAbstract:The Vafa-Witten equations on an oriented Riemannian 4- manifold are first order, non-linear equations for a pair of connection on a principle SO(3) bundle over the 4-manifold and a self-dual 2-form with values in the associated Lie algebra bundle. The main theorem in this paper characterizes in part the behavior of sequences of solutions to the Vafa-Witten equations which have no Convergent Subsequence. The paper proves that a renormalization of a Subsequence of the self-dual 2-form components converges on the complement of a closed set with Hausdorff dimension at most 2, with the limit being a harmonic 2-form with values in a real line bundle.Comment: Minor corrections for this versio
Clifford Henry Taubes - One of the best experts on this subject based on the ideXlab platform.
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The behavior of sequences of solutions to the Vafa-Witten equations
arXiv: Differential Geometry, 2017Co-Authors: Clifford Henry TaubesAbstract:The Vafa-Witten equations on an oriented Riemannian 4- manifold are first order, non-linear equations for a pair of connection on a principle SO(3) bundle over the 4-manifold and a self-dual 2-form with values in the associated Lie algebra bundle. The main theorem in this paper characterizes in part the behavior of sequences of solutions to the Vafa-Witten equations which have no Convergent Subsequence. The paper proves that a renormalization of a Subsequence of the self-dual 2-form components converges on the complement of a closed set with Hausdorff dimension at most 2, with the limit being a harmonic 2-form with values in a real line bundle.
Huseyin Cakalli - One of the best experts on this subject based on the ideXlab platform.
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A variation on arithmetic continuity
Sociedade Brasileira de Matemática, 2017Co-Authors: Huseyin CakalliAbstract:A sequence $(x_{k})$ of points in $\R$, the set of real numbers, is called \textit{arithmetically Convergent} if for each $\varepsilon > 0$ there is an integer $n$ such that for every integer $m$ we have $|x_{m} - x_{}|$ denotes the greatest common divisor of the integers $m$ and $n$. We prove that a subset of $\R$ is bounded if and only if it is arithmetically compact, where a subset $E$ of $\R$ is arithmetically compact if any sequence of point in $E$ has an arithmetically Convergent Subsequence. It turns out that the set of arithmetically continuous functions on an arithmetically compact subset of $\R$ coincides with the set of uniformly continuous functions where a function $f$ defined on a subset $E$ of $\R$ is arithmetically continuous if it preserves arithmetically Convergent sequences, i.e., $(f(x_{n})$ is arithmetically Convergent whenever $(x_{n})$ is an arithmetic Convergent sequence of points in $E$
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Sequential definitions of compactness
Applied Mathematics Letters, 2008Co-Authors: Huseyin CakalliAbstract:Abstract A subset F of a topological space is sequentially compact if any sequence x = ( x n ) of points in F has a Convergent Subsequence whose limit is in F . We say that a subset F of a topological group X is G -sequentially compact if any sequence x = ( x n ) of points in F has a Convergent Subsequence y such that G ( y ) ∈ F where G is an additive function from a subgroup of the group of all sequences of points in X . We investigate the impact of changing the definition of convergence of sequences on the structure of sequentially compactness of sets in the sense of G -sequential compactness. Sequential compactness is a special case of this generalization when G = lim .